Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations

•New functions called fractional-order Bernoulli functions are defined.•Operational matrices of the fractional integration and derivative are driven.•Our approach was based on the least square approximation method.•This method is applied for solving fractional integro-differential equations.•The convergence of the method is extensively discussed.

In this paper, we define a new set of functions called fractional-order Bernoulli functions (FBFs) to obtain the numerical solution of linear and nonlinear fractional integro-differential equations. The properties of these functions are employed to construct the operational matrix of the fractional integration. By using this matrix and the least square approximation method the fractional integro-differential equations are reduced to systems of algebraic equations which are solved through the Newton's iterative method. The convergence of the method is extensively discussed and finally, some numerical examples are shown to illustrate the efficiency and accuracy of the method.